Factorisation of Greedoid Polynomials of Rooted Digraphs

TL;DR

This paper investigates the factorization properties of greedoid polynomials associated with rooted digraphs, providing computational results, theoretical insights, and examples of factorization beyond direct sum decompositions.

Contribution

It introduces the concept of GM-factorization for greedoid polynomials of rooted digraphs and explores conditions and examples where such factorizations occur.

Findings

Computed greedoid polynomials for all rooted digraphs up to order six.

Identified conditions under which greedoid polynomials are divisible by 1 + z.

Provided examples and an infinite family of rooted digraphs with non-trivial GM-factorizations.

Abstract

Gordon and McMahon defined a two-variable greedoid polynomial $ f(G;t,z) $ for any greedoid $ G $. They studied greedoid polynomials for greedoids associated with rooted graphs and rooted digraphs. They proved that greedoid polynomials of rooted digraphs have the multiplicative direct sum property. In addition, these polynomials are divisible by $ 1 + z $ under certain conditions. We compute the greedoid polynomials for all rooted digraphs up to order six. A greedoid polynomial $ f(D) $ of a rooted digraph $ D $ of order $ n $ GM-factorises if $ f(D) = f(G) \cdot f(H) $ such that $ G $ and $ H $ are rooted digraphs of order at most $ n $ and $ f(G),f(H) \ne 1 $. We study the GM-factorability of greedoid polynomials of rooted digraphs, particularly those that are not divisible by $ 1 + z $. We give some examples and an infinite family of rooted digraphs that are not direct sums but their greedoid polynomials GM-factorise.